The alternative health chain has 750 UK stores and counts as an essential retailer because its products cater for people with special dietary needs.
It has stores in Westborough, Scarborough, Sandgate, Whitby, and the Promenades, Bridlington. The stores are, at present open, from 10am to 4pm.
But staff have set up a petition, pleading with management to close stores over safety concerns.
The chain said staff and customer safety is of "paramount importance."
The petition says: “Being on the frontlines, dealing with customers day in and day out, increases the chances of the virus spreading rapidly. The company has not been providing us hand washes, anti-bacterial gels and other form of protection to reduce the chance of catching the virus.”
The petition has garnered more than 4,000 signatures.
Holland & Barrett said staff and customer safety is of “paramount importance.”
Some staff members have turned to Twitter to voice their concerns, with one employee tweeting: “Please close your stores. 90% of customers only buy non-essential products”.
Workers have two complaints. They argue that what the chain sells is not essential, unlike supermarkets or pharmacies.
But they also warn that most stores are too small to be safe, with no room for social-distancing.
The retailer said it has provided gloves, masks and hand sanitiser to all stores and has also installed shielding screens at the counter.
It said social distancing is being enforced “with clear guidance in store.”
“We have limited the maximum number of customers allowed in the stores at any one time, dependent on store size, and tape has been installed marking out a two metre distance from the tills with clear signage throughout stores.”
A Holland & Barrett spokesperson told the BBC it had been advised by the government that it is an essential retailer expected to remain open during the lockdown.
“This relates to the fact that 90% of what we sell is food, vitamins and food supplements, which many of our customers rely on to manage their dietary or underlying conditions,” they said.
Examples the company cited included pregnant women in need of folic acid supplementation, Coeliacs, people with milk and egg allergies, those outlined by the government as “at-risk” groups in need of vitamin D supplementation, and those caring for customers with compromised immune systems, in need of additional support at this time.
Employees have highlighted that all Holland & Barrett products are available through its online store.
Holland & Barrett said: “We are bolstering our online capacity as quickly as possible so customers can order from home where possible, and have set also up a new customer phone line for those customers who cannot access our website for home delivery.”